In this paper, the mathematical structure of k orthonormalized eigenstutds of the higher powers (k≥3) of the q-boson annihilation operator are studied. Based on the work, the properties of the mathematics, q-squeezing and the amplitude N th-power squeezing are investigated. It is found that they form a complete q-Hilbert sauce, and the amplitude N th-power squeezing [N=(m+1/2)k, m=0,1,2,…] can exist in the all of them when k is even.